Subsampling estimates of the Lasso distribution.

6.2 High dimensional setting 61
3. If ˆp (1) < α/s but ˆp (2) ≥ α/(s − 1), accept H (2) , ·, H (s) and stop. If ˆp (1) < α/s and
ˆp (2) < α/(s − 1), reject H (2) in addition to H (1) and test the remaining s − 2 hypotheses
at level α/(s − 2).
One can show that the Bonferonni-Holm procedure controls the FWER (Lehmann and
Romano, 2005, Theorem 9.1.2).
In our case, for an individual hypothesis H 0,j : β j = 0, we propose the following p-value
based on subsampling: for a realized statistics √ n ˆβ j n computed on the whole sample,
choose as p-value the proportion among B subsamples of values √ r( ˆβ j n,b,i − ˆβ j n) lying
beyond or beneath it, depending thereon if it is positive or negative, more precisely:
ˆp j =
{
B
−1 ∑ B
i=1 1{ √ r( ˆβ j n,b,i − ˆβ j n) ≥ √ n ˆβ j n} if ˆβ j n ≥ 0
B −1 ∑ B
i=1 1{ √ r( ˆβ j n,b,i − ˆβ j n) ≤ √ n ˆβ j n} if ˆβ j n < 0
(6.1.3.2)
where r = b/(1 − b/n) is the finite sample corrected subsample size. Similar p-values
are considered in Berg, McMurry, and Politis (2010). Here, the disctintion between the
positive and the case is made to take in consideration power issues.
Powers and Family Wise Errors Rates based on 500 replications are reported in tables
6.4, 6.5 and 6.6. The procedure controls the FWER while leaving the power unaffected,
indicating that the proposed p-values are correct.
6.2 High dimensional setting
To access the performance of subsampling in a high dimensional setting, we apply it to the
adaptive Lasso presented in Chapter 3. We generate four data sets D,D’,E and E’ which
differ in the correlation structure between covariates and the dimension of the parameters.
They result from the linear model
where
Y i = x ′ iβ + ε i , i = 1, . . . , n
β = (1, −1.25, , −1.25, 1, −1.5, 0) ′
is a 400 × 1 vector in models D and D’, and a 800 × 1 vector in models E E’. The errors ε i
are taken i.i.d. standard normal. X is multivariate normal with mean zero and covariance
matrix C given by
• C i,j = 0.8 |i−j| 1{i, j ∈ {1, . . . , 5} or i, j ∈ {6, . . . , 400}} in model D,
• C i,j = 0.8 |i−j| , i, j = 1, . . . , 400 in model D’,
• C i,j = 0.8 |i−j| 1{i, j ∈ {1, . . . , 5} or i, j ∈ {6, . . . , 800}} in model E,